Growth of bilayer MoTe2 single crystals with strong non-linear Hall effect

The reduced symmetry in strong spin-orbit coupling materials such as transition metal ditellurides (TMDTs) gives rise to non-trivial topology, unique spin texture, and large charge-to-spin conversion efficiencies. Bilayer TMDTs are non-centrosymmetric and have unique topological properties compared to monolayer or trilayer, but a controllable way to prepare bilayer MoTe2 crystal has not been achieved to date. Herein, we achieve the layer-by-layer growth of large-area bilayer and trilayer 1T′ MoTe2 single crystals and centimetre-scale films by a two-stage chemical vapor deposition process. The as-grown bilayer MoTe2 shows out-of-plane ferroelectric polarization, whereas the monolayer and trilayer crystals are non-polar. In addition, we observed large in-plane nonlinear Hall (NLH) effect for the bilayer and trilayer Td phase MoTe2 under time reversal-symmetric conditions, while these vanish for thicker layers. For a fixed input current, bilayer Td MoTe2 produces the largest second harmonic output voltage among the thicker crystals tested. Our work therefore highlights the importance of thickness-dependent Berry curvature effects in TMDTs that are underscored by the ability to grow thickness-precise layers.

nm. As the CVD growth process is near equilibrium state, the zigzag chain direction is the most energy favourable edge 1 . The monoclinic crystal structure causes the morphology of the 1T′ MoTe2 to be rectangular, and this has also been commonly observed 2-4 . (006), (008) planes of the MoTe2 crystals, which is in good agreement with the reference pattern for monoclinic 1T′ phase. Obviously, all prominent diffraction peaks are indexed to the {001} family planes, suggesting that the c-axis of the as-grown film is perpendicular to the growth substrate and that the growth is highly textured. for normal geometry (black) and opposite geometry (blue), respectively. In the main text, we focus on the 2 nd harmonic Hall signal where the current is applied along a-axis and we detected the voltage difference along b-axis (as denoted in the inset a). This measurement setup is inspired by the theory that the Berry curvature dipole lies along a-axis due to mirror symmetry in MoTe2. Here, we also measured the 2 nd harmonic Hall signal in the opposite geometry. In this opposite geometry, the current passes along b-axis and we detected the voltage along a-axis (as denoted in the inset b). When plotted together, we can find the 2 nd harmonic response acquired in opposite geometry is much weaker. This further confirms the symmetry selection rule.  MoTe2 shows clearly the characteristics of 1T′ phase, which is also in agreement with the STEM observations. For the layer numbers, the higher energy mode at ≈ 271 cm -1 red shifts when the thickness of 1T′ MoTe2 increase from monolayer to bilayer (268 cm -1 ), and trilayer (265 cm -1 ). This is because the out-of-plane Ag mode at this energy mode is strongly affected by the interlayer interactions. It is considered that such a frequency drop with increased thickness is possibly due to the enhancement of dielectric screening of the long-range Coulomb interaction in thicker MoTe2 6 . Therefore, a fingerprint Raman peak at 269 cm -1 distinguishes monolayer MoTe2 and the peak position redshifts to 267, and 265 cm -1 for bilayer and trilayer, respectively. As shown in Supplementary Figure  The Berry curvature distributions were calculated in two different schemes, fixed-Fermi-level scheme ( Fig. 3c and e) and fixed-number-of-occupied-bands scheme ( Fig. 3d and f) It is noteworthy that the crossing points of the Td-bilayer are not the Weyl points, which is reported in the bulk MoTe2 19 . Weyl points always exist as a pair of a source and a sink of the Berry curvature. However, in the bilayer, the two crossing points represent a pair of two Berry curvature sources.

Supplementary
In our case, peaks in the Berry curvature dipole are far below the Fermi level due to doping.
Therefore, the corresponding Berry curvature dipole is much smaller than the peak values.
However, one should note that the peak level may depend also on the details of the calculation, e.g., the HSE parameter.

Supplementary Note 3. Device fabrication and encapsulation
The as-grown MoTe2 on SiO2/Si wafers are instantly transferred into argon-filled glovebox after CVD growth. Large rectangular bilayer (trilayer) MoTe2 flakes are examined under optical microscope. E-beam resist is subsequently spin-coated and sample is pre-exposure baked. Standard e-beam lithography process and e-beam evaporation process is done. After that sample is transferred into the glovebox for liftoff, a thin (20-40 nm) hexagonal boron nitride flake is transferred on top to encapsulate sample and prevent degradation.

Supplementary Note 4: Electrical characteristics of bilayer MoTe 2
A rectangular bilayer MoTe2 Hall bar sample has only crystalline symmetry which is the mirror plane Ma. Thus an AC current along a-axis would cause a 2 nd harmonic Hall response along baxis. Immediately after fabrication, the device was loaded in the Oxford Teslatron system. AC measurements were conducted with lock-in instruments (SR830). All measurements were conducted at base temperature 1.6 K unless otherwise stated. During cooling down, the resistance of bilayer MoTe2 shows an upswing at low temperatures.
The upswing in resistance at low temperature for few-layer samples has been reported previously [20][21][22] . The transport behavior of WTe2 goes through metal-insulator transition when the vertical thickness of the sample is reduced 20 . The authors found that the mobility of the charge carriers are 2-3 orders of magnitude smaller than that of thicker layers and thus disorder should be the cause. They argue that the carriers are Anderson localized. The metal-insulator transition in few-layer MoTe2 is also explained as enhanced charge carrier localization 21 .
We have also noticed a report that even monolayer MoTe2 shows metallic behavior down to low temperature 23 . The main difference is that the authors used hBN to fully encapsulate the MoTe2 flake. In our case, our sample was only covered by hBN on top to provide protection from degradation and the SiO2/Si substrate has an influence on MoTe2. In surveying literature, we found that the substrate influences the device behavior of the MoTe2 sample significantly, in which few-layer MoTe2 has been reported to exhibit insulating or 'gapped/band-splitting' behaviors depending on the substrate. For instance, the R-T relation of the 2 nm-thick MoTe2 (can be considered as bilayer) also shows an upswing below 50 K before transition into superconducting state 22 . We observed that the authors used SiO2/Si substrate similar to ours.
Additionally, monolayer MoTe2 exhibit semi-metallic behavior with large band overlap when grown on bilayer graphene 24 but a weak overlap with a potential gap-opening when exfoliated on gold substrates 25 . Therefore, we can infer that the electronic band structure of few-layer MoTe2 is very sensitive to the substrate. Based on the above analysis, our bilayer sample may have more skew scattering from SiO2/Si substrate compared to hBN. This point is also reflected in our discussion of the NLH magnitude in the paper in which we stated that the NLE has multiple contributions (i.e., skew scattering etc), and not limited to intrinsic Berry curvature.
Due to low conductivity of the bilayer, the relation of NLH magnitude with conductivity is complex and cannot be quantified analytically at the present stage. In contrast, for metallic trilayer MoTe2, the linear relation in Fig. 5d clearly agrees well with both theory and published literatures.
Apart from the 2nd order Hall response, we have also measured the 2nd order longitudinal response. A comparison of these two responses plotted against longitudinal electric field is shown in Supplementary Figure 19. It is clearly shown that the 2nd order Hall signal dominates over the longitudinal counterpart, which also rules out the possibility of thermal-induced 2nd order effect which is isotropic. Also, we observed that the phase of 2 nd Hall voltage with respect to longitudinal voltage has a 90-degree phase shift, as shown in Supplementary Figure 20. The consistent phase further supports the 2 nd order origin since a straightforward relation of 2 nd harmonic signal should follow as: Where the 2 nd order part should have a 90-degree phase shift with respect to applied voltage.

Supplementary Note 5. Electrical characteristics of trilayer MoTe 2
Trilayer CVD-grown MoTe2 devices are also fabricated and measured in much similar approach to provide a contrastive counterpart. In general, trilayer samples show metallic behavior in resistivity-T measurement (Supplementary Figure 22). This metallic behavior is consistent with previous reports 26,27 , which consolidates the quality of our grown samples.
For Fig. 5d in the main text, we can observe linear dependence relation in thicker samples 8,15 .
We can also rewrite the NLH magnitude in terms of anomalous/longitudinal conductivity ratio is the anisotropic resistance ratio of around 0.37 from our experiment.
In low-frequency limit, as in our experiment, the intrinsic contribution to the nonlinear Hall conductivity 28 can be written as = 2 0 , where D is Berry curvature dipole, 0 is the conductance quantum and ℏ ≡ ∥ is the net electron momentum obtained under an inplane bias ∥ (ℏ and e denote, Planck constant electron charge). The longitudinal conductivity can be expressed as = 0 F F /2 , where F and F are the Fermi velocity and Fermi vector, respectively, and are related to the Fermi energy as ~ ℏ F F . We can thus obtain AH ||~ℏ F F~F / .
In this relation, the dipole can be evaluated from the experimental value of AH || in the limit of → 0 where the only contribution comes from intrinsic effect Berry curvature dipole D (neglect side jump). Thus, a linear fit to curve in Fig. 5d can yield y-axis intercept ~1.34 * 10 −2 μm V −1 , which scales directly with Berry curvature dipole if we neglect the contribution from side jump.
This value is comparable to values reported for bilayer WTe2 recently 7 , but one order of magnitude larger than that of few-layer WTe2 8 .